Wold-type decompositions and wandering subspaces for operators close to isometries (Q2703581)

For a bounded operator \(T\) on a complex Hilbert space \(H\), denote \(H_{\infty}=\bigcap_{n\geq 1}T^n H\), \(E=\text{Ker} T^*\) and \([E]_T\) the smallest closed \(T\)-invariant subspace of \(H\) containing \(E\). In the case when \(T\) is an isometry, the classical Wold decomposition is \(H=H_{\infty}\oplus[E]_T\), this being a decomposition of \(H\) into a direct sum of two closed \(T\)-invariant subspaces, where \(T\) restricted to \(H_{\infty}\) is unitary while \(T\) restricted to \([E]_T\) is a shift. Moreover \(E\) is a wandering subspace in the sense that \(E\bot T^nE\) for every \(n\geq 1\). NEWLINENEWLINENEWLINEThe paper under review investigates various versions of these classical facts, in the case of operators \(T\) which are ``close to isometries'', being in particular left invertible. One of the main theorems says that, if \(T\) satisfies either \(\|T^2 x\|^2+\|x\|^2\leq 2\|Tx\|^2\) for every \(x\in H\), or \(\|Tx+y\|^2\leq 2(\|x\|^2+\|Ty\|^2)\) for all \(x,y\in H\), then \(H_{\infty}\) is reducing for \(T\), \(T|_{H_{\infty}}\) is unitary and \(H=H_{\infty}\oplus[E]_T\). The connection between the above inequalities is pointed out by means of the operator \(T(T^*T)^{-1}\), which turns out to be in some sense dual to \(T\), thus finally establishing some type of duality between those inequalities. NEWLINENEWLINENEWLINEThe abstract results of the paper are applied to shift-invariant subspaces in weighted Bergman spaces. This application is motivated by the results of the important paper by \textit{A. Aleman, S. Richter} and \textit{C. Sundberg}, ``Beurling's theorem for the Bergman space'', Acta Math. 177, No. 2, 275-310 (1996; Zbl 0886.30026).